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We give a criterion for cohomological symmetry in a triangulated category. As an application, we show that such cohomological symmetry holds for all pairs of modules over any exterior algebra.

With the couplings between the eight gluons constrained by the structure constants of the su(3) algebra in QCD, one would expect that there should exist a special basis (or set of bases) for the algebra wherein, unlike in a Cartan-Weyl…

We realize the enveloping algebra of the positive part of a semisimple complex Lie algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.

In this work, we explore the close relationship between an ideal map structure S --> End(R) on a homomorphism of commutative k-algebras R --> S and an ideal simplicial algebra structure on the associated bar construction Bar(S, R).

Let ${\mathcal N}$ be the Lie algebra of all $n\times n$ strictly block upper triangular matrices over a field ${\mathbb F}$ relative to a given partition. In this paper, we give an explicit description of all derivations of ${\mathcal N}$.

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a…

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both…

Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…

Let $UT_2$ be the algebra of $2\times 2$ upper triangular matrices over a field $F$ of characteristic zero. Here we study the generalized polynomial identities of $UT_2$, i.e., identical relations holding for $UT_2$ regarded as…

In this note, we study the determinantal structure of the $k$-th conditional expectation of the overlap for induced spherical unitary ensemble. We will show the universality for the scaling limits of the $k$-the conditional expectation of…

We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all…

We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3\times 3$ upper triangular matrix algebra over an infinite field.

In this document, some structured operator approximation theoretical methods for system identification of nearly eventually periodic systems, are presented. Let $\mathbb{C}^{n\times m}$ denote the algebra of $n\times m$ complex matrices.…

The goal of the paper is to give a complete description of the images of noncommutative polynomials with zero constant term on upper triangular matrix algebras over an algebraically closed field. This is a variation of the old and famous…

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some…

In this paper we define infinite-dimensional algebra and its representation, whose basis is naturally identified with semi-infinite configurations of the square ladder model. We also extrapolate the ideas for the cyclic 3-leg triangular…

A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated…

We propose a correspondence between vertex operator superalgebras and families of sigma models in which the two structures are related by symmetry properties and a certain reflection procedure. The existence of such a correspondence is…

Describing the conjugacy classes of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ uniformly (for all or many values of $n$ and $q$) is a nearly impossible task. This paper takes on the related problem of describing…

We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between super-integrability and exact solvability is illustrated. Eigenfunctions are…